23,645 research outputs found
Measurable events indexed by trees
A tree is said to be homogeneous if it is uniquely rooted and there
exists an integer , called the branching number of , such that
every has exactly immediate successors. We study the behavior of
measurable events in probability spaces indexed by homogeneous trees.
Precisely, we show that for every integer and every integer there exists an integer with the following property. If is a
homogeneous tree with branching number and is a family of
measurable events in a probability space satisfying
for every , then for every
there exists a strong subtree of of infinite height such that for every
non-empty finite subset of of cardinality we have
\mu\Big(\bigcap_{t\in F} A_t\Big) \meg \theta^{q(b,n)}. In fact, we can take
. A finite version of this
result is also obtained.Comment: 37 page
Optimal investment under multiple defaults risk: A BSDE-decomposition approach
We study an optimal investment problem under contagion risk in a financial
model subject to multiple jumps and defaults. The global market information is
formulated as a progressive enlargement of a default-free Brownian filtration,
and the dependence of default times is modeled by a conditional density
hypothesis. In this Ito-jump process model, we give a decomposition of the
corresponding stochastic control problem into stochastic control problems in
the default-free filtration, which are determined in a backward induction. The
dynamic programming method leads to a backward recursive system of quadratic
backward stochastic differential equations (BSDEs) in Brownian filtration, and
our main result proves, under fairly general conditions, the existence and
uniqueness of a solution to this system, which characterizes explicitly the
value function and optimal strategies to the optimal investment problem. We
illustrate our solutions approach with some numerical tests emphasizing the
impact of default intensities, loss or gain at defaults and correlation between
assets. Beyond the financial problem, our decomposition approach provides a new
perspective for solving quadratic BSDEs with a finite number of jumps.Comment: Published in at http://dx.doi.org/10.1214/11-AAP829 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management
We formulate and investigate a general stochastic control problem under a
progressive enlargement of filtration. The global information is enlarged from
a reference filtration and the knowledge of multiple random times together with
associated marks when they occur. By working under a density hypothesis on the
conditional joint distribution of the random times and marks, we prove a
decomposition of the original stochastic control problem under the global
filtration into classical stochastic control problems under the reference
filtration, which are determined in a finite backward induction. Our method
revisits and extends in particular stochastic control of diffusion processes
with finite number of jumps. This study is motivated by optimization problems
arising in default risk management, and we provide applications of our
decomposition result for the indifference pricing of defaultable claims, and
the optimal investment under bilateral counterparty risk. The solutions are
expressed in terms of BSDEs involving only Brownian filtration, and remarkably
without jump terms coming from the default times and marks in the global
filtration
Local Tomography of Large Networks under the Low-Observability Regime
This article studies the problem of reconstructing the topology of a network
of interacting agents via observations of the state-evolution of the agents. We
focus on the large-scale network setting with the additional constraint of
observations, where only a small fraction of the agents can be
feasibly observed. The goal is to infer the underlying subnetwork of
interactions and we refer to this problem as . In order to
study the large-scale setting, we adopt a proper stochastic formulation where
the unobserved part of the network is modeled as an Erd\"{o}s-R\'enyi random
graph, while the observable subnetwork is left arbitrary. The main result of
this work is establishing that, under this setting, local tomography is
actually possible with high probability, provided that certain conditions on
the network model are met (such as stability and symmetry of the network
combination matrix). Remarkably, such conclusion is established under the
- , where the cardinality of the observable
subnetwork is fixed, while the size of the overall network scales to infinity.Comment: To appear in IEEE Transactions on Information Theor
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